Question

For complexes of \(\mathrm{V}^{3+}\), only one distribution of electrons is possible. Explain.

Short answer

Answer: There is only one possible distribution of electrons for complexes of V³⁺ due to its d² electron configuration. Both of these electrons will occupy the lower-energy t₂g orbitals, and there will be none in the e_g orbitals. This electron distribution remains the same regardless of the geometry and ligand field strength of the complex.

Step-by-step solution

Determine the electronic configuration of V³⁺

Vanadium (V) is element number 23 on the periodic table, which means it has 23 electrons in its neutral state. Its electron configuration is [Ar] 4s² 3d³. In the V³⁺ ion, three electrons are removed, leaving it with 20 electrons. Since the 4s orbitals are filled before the 3d orbitals, the electrons will be removed from the 4s orbitals first and then from the 3d orbitals. So, the V³⁺ ion will have the electron configuration: [Ar] 3d².

Consider the different orbital splitting scenarios

In a complex, electron distribution can be affected by the splitting of d orbitals into two groups with different energy levels: t₂g (lower energy) and e_g (higher energy). This splitting is determined by the ligands surrounding the central metal ion and the geometry of the complex. In the case of V³⁺, since we have a d² configuration, both of these electrons will occupy the t₂g orbitals (lower energy) following Hund's rule, which states that electrons will occupy degenerate orbitals singly before pairing up. Regardless of the geometry and the ligand field strength, only one electron distribution is possible for V³⁺ because there are not enough electrons in 3d orbitals to create varied distributions. So, for any complex of V³⁺, the distribution of electrons will always be two electrons in the t₂g orbitals and none in the e_g orbitals.

Key concepts

Understanding the Periodic Table

The periodic table is the cornerstone of chemistry, organizing all known elements by their atomic number or the number of protons in their nucleus. For students delving into complex ions, the periodic table becomes essential for determining electron configurations, which is the starting point in this exercise. Vanadium (V), with an atomic number of 23, has its electrons arranged according to the table's structure. Initially, vanadium has the electron configuration [Ar] 4s² 3d³, based on its position among transition metals.

When we consider the V³⁺ cation, we refer to the periodic table to see that electrons are first removed from the outermost shell. The table also delineates the order of electron filling across the s, p, d, and f orbitals, which comes into play when these elements form ions and later, complex ions. It's critical to understand where an element lies on the periodic table to predict its electron configuration and subsequent behavior in compounds.

Orbital Splitting in Complex Ions

When metal ions form complexes, the energy levels of their d-orbitals change due to the influence of the surrounding ligands. This phenomenon is known as orbital splitting and is explained by crystal field theory. In simpler terms, the presence of ligands around a central metal ion causes the five degenerate d-orbitals to split into two groups with different energy levels: typically, a lower-energy triplet set called 't₂g' and a higher-energy doublet set named 'e_g'.

For the V³⁺ ion in this exercise, which has a d² electron configuration, orbital splitting is uncomplicated. Regardless of the geometry of the complex (whether it is octahedral, tetrahedral, or another shape), the two electrons of V³⁺ will always occupy the t₂g orbitals following the path of least resistance to lower energy states. Understanding this splitting is crucial for anyone studying coordination chemistry as it impacts color, magnetic properties, and the reactivity of complex ions.

Applying Hund's Rule to Electron Configurations

Hund's rule is vital when assessing electron distribution in orbitals, particularly in scenarios with multiple orbitals of the same energy level, known as degenerate orbitals. The rule states that electrons will fill degenerate orbitals singly as far as possible before pairing up. This minimizes repulsion between electrons and consequently, the energy of the atom or ion.

In the case of V³⁺, with a 3d² electron configuration, Hund's rule dictates that each of the two electrons resides in separate t₂g orbitals rather than pairing in one orbital. It's based on the idea that electrons, being negatively charged, repel each other and prefer to stay as far apart as possible. This underlines why, regardless of the complex's exact geometry or the ligands present, V³⁺ ion can only have one electron arrangement. Hund's rule simplifies the prediction of electron distribution, serving as a guide in determining the magnetic and spectral properties of complexes.